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Theorem rexcom13 3081
 Description: Swap first and third restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
Assertion
Ref Expression
rexcom13 (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
Distinct variable groups:   𝑦,𝑧,𝐴   𝑥,𝑧,𝐵   𝑥,𝑦,𝐶
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑧)

Proof of Theorem rexcom13
StepHypRef Expression
1 rexcom 3080 . 2 (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑦𝐵𝑥𝐴𝑧𝐶 𝜑)
2 rexcom 3080 . . 3 (∃𝑥𝐴𝑧𝐶 𝜑 ↔ ∃𝑧𝐶𝑥𝐴 𝜑)
32rexbii 3023 . 2 (∃𝑦𝐵𝑥𝐴𝑧𝐶 𝜑 ↔ ∃𝑦𝐵𝑧𝐶𝑥𝐴 𝜑)
4 rexcom 3080 . 2 (∃𝑦𝐵𝑧𝐶𝑥𝐴 𝜑 ↔ ∃𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
51, 3, 43bitri 285 1 (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902 This theorem is referenced by:  rexrot4  3082
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